C H A P T E R ~ : T w o - d i m e n s i o n a l d o m a i n s

§13 G e n e r a l i t i e s about two-dimensional domains

13.A Introduction

We are going to precise in 13.C h o w to d e t e r m i n e w h e t h e r an o p e r a t o r is in j ec t ive

modulo po lynomia l s on S~(I ") w h e n 1" is a p lane sector . But, b e f o r e h a n d , w e are going to

p r e sen t in 13.B a n o t h e r p roof in t h e case of l imit e x p o n e n t s for a p l ane crack. That new

proof is u n n e c e s s a r y s ince w e had just p roved t h a t resul t in §~ 9 and 10. But t h a t al lows to

s tudy Kondra t ' ev ' s me thod , w h i c h is in t e res t ing .

For t w o - d i m e n s i o n a l domains , t he model domains are all cones I '~@2.

We d e n o t e Y° :=R2 \{0} , t he model for a hole ; and for each ~0~]0,2~], we d e n o t e I " the

p lane sec tor :

F % { z E R 2 / z/lzl ~ ]0,~0D.

If ~0=2~, it is t he model of a crack.

If co = ~, i t is t h e model of a smooth boundary .

The set e 2 consists o f all F° for o~ E [0,2E]. - cf example (2 .9 /3) .

We d e n o t e ~ the sect ion of Y~.

• If 0J=0, ~ is equal to S I .

o If o~]0,2Tt[ , ~ ~ ]0,0~[ and (~o)R ~ [0,C0] - cf (2.11).

o I fc0=2~, Q°=SJX{I} and (~)R=SJ .

In a n y case, A(~ ~) is empty , i.e. (fi%R is a smooth mani fo ld ( w i t h o u t b o u n d a r y if 0J=0, 2E,

and w i t h b o u n d a r y if co ~ ]0,2~t[).

Now let L be an h o m o g e n e o u s 2 m - o r d e r o p e r a t o r w i t h c o n s t a n t coeff ic ients . ~ is

still de f ined as :

2 m r L(D z) = ~ ( 0 ; r 0 r , D O )

And ~ ( ~ ) is sti l l t h e o p e r a t o r ~_,(0 ; }., Do). As A(O °) = ~, w e have , for a n y s and any

to ~ [0,2~] :

V ; t E C , ~ ( ; t ) is [O ,s ] - regu la r on I~ra(fl~) .

Then , l emma (4.4 ') i n fe r s t h a t for each s~>0, t h e fo l l owing c o n d i t i o n s are e q u i v a l e n t to

105

each o t h e r :

( 1 3 . 1 ) L is in j ec t ive modulo po lynomia ls on S~(£ ~)

( 1 3 . 2 ) L is in j ec t ive modulo po lynomia l s on s~ ' s (F°) .

Thus, it is e n o u g h to s tudy (13.1).

13.B Limit exponents for the plane c r ac k

We are go ing to p r o v e t h e f o l l o w i n g resu l t by K o n d r a t ' e v ' s m e t h o d of so lv ing

$ modulo H o .

(13.3) L e m m a : L e t L be a p r o p e r e l l ip t ic , h o m o g e n e o u s 2 m - o r d e r o p e r a t o r

cons tant coef f ic ients . Le t s E N , with s >1 m + 1. We assume that :

V~,, Re Z=s+:m-1, L is injective modulo po lynomia l s on S~(F2~).

Let ~ > 0 be smal l enough to keep that proper ty on the strip R e ~ , 6 [ s + m - l - E , s + m - 1 ] .

Then L is [ s = ~ , s ] - r e g u l a r in 0 on I~Im(F2~).

with

To p rove (13.3), it is enough to s ta te t he fo l lowing theo rem. Let us in t roduce for s e N and

£ E ~ 2 = cf (AA.7) :

H s ( r ) = (uEHS(r) / Dau(0)=0,1al~<s-2}.

(13.4) T h e o r e m : W e suppose that :

f o r Z = s + m = l , L is injective modulo po l ynomia l s on S~(£ zn ).

Then, f o r any f E H s,- m ( £ 2 ~ ), there exists u 0 belonging to H s,+ m n ~I m (£ 2 ~ ) such that :

_ s - m (F2~ Lu o f e H o ).

S k e t c h of p r o o f o f (13 .3 ) :

Let u EHs+m-~NI~ m be such t h a t LuEH s-re . We suppose t h a t E<I. By subs t rac t ing

f rom u its degree ( s + m - 2 ) Taylor expans ion in 0 we reduce to the case w h e n :

u E H s+ m - c h U m is such t h a t LueHS, -m

By subs t rac t ing f rom u t h e f u n c t i o n u o of (13.4) associated to f=Lu, w e reduce to the

case w h e n :

u e H S , + m - t n ~ m is such t h a t LuEH o-m

Then , w e p r o v e d i r e c t l y t h a t for a n y It~i=s+m, D~u be longs to L 2 t h a n k s to an

106

a r g u m e n t of "regular pole", i.e. a pole associated to a po lynomia l residue cf [KO I]. •

R e m a r k : Our g e n e r a l m e t h o d is i n sp i r ed by t h a t idea of r egu la r pole : i n s t e ad of

s - m s - m s tudy ing r igh t h a n d sides in H o w e are able to s tudy d i rec t ly r igh t h a n d sides in H,

The proof of(13.4) is based upon t he two fo l lowing resul ts : t h e f i rs t one, (13.5), is a

cons equence of Hardy inequa l i t y and t h e second, (13.6), is a s t r a i g h t f o r w a r d consequence

of lemma 4.17 in [KO 1] w h e n n=2 .

(13.5) L e m m a : Let F c e n, Let s be such that s - n / 2 a N .

n / 2 t f \v eHS, (F) is such that for each ~, I ~ l = s - n / 2 we have : D a W a H o (£), then :

w e l l S ( F ) .

Tha t lemma is a c o m p l e m e n t of (AA.7).

(13.6) L e m m a : Let F e e z. Let k e n . W e denote ~ k

( k - 1), differential operators with constant coefficients.

Let G 1 ..... G i be linearly independant elements of ffe,~.

Let fv...,fi be elements of H t (F).

Then, there exists f e H ~ , (F) such that :

1 H i f - f i c H o ( r ) , Vi=l,...,j.

Proof of (13.4) : We d e n o t e (x,y) t he coord ina tes in R 2 such t h a t £ z~ =R z \{x,0)/x~> 0}.

We wi l l p r o v e t he ex is tence of v e i l s , + m(F~) such t h a t :

et I D ( L v = f ) e H o ( F ) , ~'c(, ~ l = s - m - 1 (I)

Dx + ra- 1 - j j I DyV E Ho(F), Vj, j=0,. . . ,m=l (2)

In a f i rs t stage, w e deduce (13.4) f rom the ex is tence of v.

(1) joined w i t h (13.5) yields :

Lv_fEHSo - m (3)

. ~1 v Let wj- be t he t races of ~ on each of t he t w o sides of £ 2~ ( c o r r e s p o n d i n g to 0=0 et

a n

0=2tO. Thanks to (2) w e get :

DS+ m - l - j _* 1/2 wi e H o (R +) (4)

the space of homogeneous degree

107

1 1 2 + (we may prove that the trace acts from H1o(I ") into H o ( R ) thanks to theorem (AA,3)).

(4) joined w i t h (135) yields :

_+ /2(R + ). Wj ~H~ ÷ m- j - I

Thanks to theorem (AA.3) we prove that there exists WCHo + m (F2~) whose m first traces +

on O'F 2" are the wj-. By set t ing Uo:=V=W w e get the conclusion of (13.4) thanks to (3).

To state the exis tence of v sat isfying (I) and (2), we def ine ~ as the subspace of

ff~s+m - cf (13.6) - spanned by the operators D~oL for l~ l=s -m-1 , and by D: + m- i - I DyJ

for j=0,...,m-1. Considering lemma (13.6), it is enough to p rove that the dimension of ~ is

s - m + m = s . The dual space of ff~s+m is the polynomials space QS+ m-1. To get tha t d im~=s ,

as the dimension of ~ s + m is s+m, it is enough to state :

dim ~ ±= m

w h e r e 2 ± is the space span by the polynomials PeQS+ m-1 such tha t for any G e ~ , ~P=0.

So, let P c ~ . We have :

D~(LP)= 0 VcL Ic4=s+m- 1.

As LP~Q s -m- l , we deduce that

LP=0. (5)

On the o ther hand :

DxS + m- j - t DyJ P = O j= O,...,m - 1.

In the same way , we get :

J Dy Ply-o = 0. (6)

(5) and (6) yield :

~ = {PEPS+ m - I ( £ 2 ~ ) / LP = 0}.

As L is in jec t ive modulo polynomials on S~(F 2~) w i t h 3 .=s+m-I then L is sur ject ive from

P~(£ 2~) onto Q~-2 m(i.2~ ). Therefore :

dim ~.L= dim P~(F 2~) - dim Q~'-am(Fa~t) = s - ( s=m)=m. •

13.C Injectivity modulo polynomials

We h a v e seen in §4 tha t the in jec t iv i ty modulo polynomials is l inked to the spectrum

of the fami ly (~(3.))~ and to the degree a of the gene ra to r A of the ideal I(I") of the

polynomials wh ich are zero on the boundary of £.

• If 00#O,x,2x, then a=2 : as a consequence of (4.9) the in jec t iv i ty modulo polynomials on

108

( 1 3 . 9 )

We set :

S~(F ~) is equ iva len t to :

X~Sp(~_,).

• If co=2~t, t hen a = l and if (0=0, a=0. We are in the scope of (4.10). See also (4.16/0 and I).

Anyway , we h a v e to know S p ( ~ ) and to study precisely the in teger poles of ~,(~t) -I

(see §~ 14, 15 for that last point).

Now, we recal l the method of [DA 2] for the calculus of the spectrum -see also

[MA-PL 5].

We wi l l apply tha t method to second order opera tors in ~14 and to four th order

operators on a crack (]7 za) or a hole (F °) in § 15.

We w r i t e the coordinates in R ~ on the form z=(x,y) , and w e take advan tage tha t L

may be factor ized in the fo l lowing w a y :

( 1 3 . 7 ) L(Dx,Dy) = H l<j<2m (Dy-aiDx)

w i t h a j~C ( that fact is specific to dimension 2). Thanks to the proper e l l ip t ic i ty of L we

get tha t (wi th possible renumber ing) :

( 1 3 . 8 ) Im aj>0 if j=1,...,m ; I m aj<0 if j=m+1,...,2m.

For ~ C , let ~ ( ~ ) be the space of solutions Z of ~_,(),)Z=0 belonging to :

, Hm(]0,~0D w h e n c0~]0,2~]

• Hm(]0,2~[) w h e n ~0=0.

The dimension of ~ ( Z ) is equal to 2m.

For any f ixed ~.oCC, the re exists a ne ighborhood cV(X o) of )'o in C such that there

exists for any ), ~ cbe(),o), a basis Zj().),...,Z2m(),) of ~(}.), the Zj being holomorphic on ave(),o).

The fo l lowing funct ions wil l be useful to calculate the Zj:

0 ~o • c o s 0 - s i n O

Fj(0) = fi(~)d~ where f i (O) - J & sinO* COSO !

( 1 3 . 1 0 ) Kj(~)(e) = e J

When the a i are dis t inct f rom each other , we may take Zj()~)=Kj(~.)'except on a f in i te

subset of C.

For ~o~C, let d)n(~L) be the map from ~() , ) into C T M which associates to Z ~ ( ~ ) the

boundary values of Z according to :

109

0 m - ! 0 m - ! • i f 6 0 x O : (Z( ;O(O) . . . . . 0 Z ( ~ . ) ( ) , Z ( ~ ) ( t 0 ) . . . . . O 0 Z(~ , ) ( t0 ) )

2m-I 2m-1 , if 6o=0 : (Z(Z)(2~t) = Z(I ) (0) . . . . . 0 e Z(2t)(2x)-~ o Z().)(0)).

I t is poss ib le t o p r o v e t h a t :

(13.11) L e m m a : W e have :

( 1 ) k / ; ~ C Ker ~£,(~.) = Ker ~]rO,)

( 2 ) I f 2t i s such that Ker J1P() ,)~{0}, then :

Pol p_~ ~g..(l~) -t = Pol p . ~ dleO,) -t o T

where T is a surjective map f r o m H - m (f2 ~) onto C z m

(We denote Po l the polar par t o f a meromorphic funct ion)

So, i t is pos s ib l e to k n o w t h e po la r p a r t o f ~f.,(~()-~ as soon as w e k n o w a basis Zj(;() of

~ ( 2 , ) . In t h e v i e w of l e m m a s (4 .9) a n d (4.10), i t is e n o u g h to s t u d y dle(~.) in o r d e r to

d e t e r m i n e t h e i n j e c t i v i t y m o d u l o p o l y n o m i a l s .

§ 1 4 S e c o n d o r d e r operators

L is s t i l l supposed to be h o m o g e n e o u s w i t h c o n s t a n t c o e f f i c i e n t s a n d w e w r i t e i t on

t h e f o r m ( I3 .7 ) :

L = ( D y - a 1 D x ) ( D y - a 2 Dx).

Here a~ a n d a 2 a r e d i s t i nc t s , fo r , a c c o r d i n g to ( I3 .8) :

Im a~ := [~j> 0 a n d Im a 2 := -[32< 0.

If a 2 = ~ ) , L i s r e a l . If a j := i a n d a 2 = - i , L = - A .

14.A Spectrum of the associated holomorphic family

The w r o n s k i a n of t h e Ki(~) - cf (13.10) et (13.9) = is equa l to ( w h e n c o m p u t e d in 0 = 0 ) :

~ ( a a - a 2 ) .

So, e x c e p t w h e n ~ = 0 , w e m a y t a k e :

Zi(),) = KjUt) j=1,2.

For c0~]0 ,2~] , t h e f u n c t i o n oW(;t.) in (13.11) has t h e f o l l o w i n g m a t r i x in t h e bas is (Zi(~)) of

1 1

~F (~) ~F (o) e e

I t s d e t e r m i n a n t is zero i f and 0 n l y if :

( 1 4 . 1 ) sh J.(F2(0~)-F~(~0))/2 = 0

i.e. if J. is equa l to o n e of t h e lk :

( 1 4 . 2 ) ;~: = 2ikr~(Fz(0~ ) - Fl(a))) -1 , k ~ Z .

By t h e loca l s t u d y in t h e n e i g h b o r h o o d of 1 = 0 it is poss ib le to p r o v e t h a t 0~Sp(~£,). So :

Sp(~..) = ( ~ / k e Z ~ ) .

If L = A , w e h a v e : f~=i, f2= - i in (13.9), a n d t h u s :

;t.~ =krt/aL k ~ Z *

as i t is w e l l k n o w n .

W h e n ~ ] 0 , 2 ~ [ , a n d ~ . ~ t , w e t h e n ge t t h a t L is i n j e c t i v e m o d u l o p o l y n o m i a l s on

S~(F ~) ff a n d o n l y if :

111

) .~ (kk / k ~ Z }.

We wi l l s tudy in sec t ion 14.B the case of t he crack ( o = 2 ~ ) . For second order

operators, a hole is i r re levant , for ~I j (R 2 \{0}) cofncides w i t h H I (R z) = cf (3.1).

Now, we are going to give informat ions about the supremum 11(~o) of the s such that

the strip Re~,~[0,s] is f ree of e lements of Sp(YJ). For such an s, L is [0,s]-regular in 0 on

~ ' (F~) . (14.2), infers that :

4(60 ) = 2r~lIm(Fz(~0 ) - FI(o )) -11.

When L=A, q ( o ) = ~ / o . So, for the Laplace operator , the func t i on 1] en joys the two

fo l lowing proper t ies :

( 1 4 . 3 ) rl((o) ~+oo w h e n ~o 4 0 ,

( 1 4 . 4 ) q is s t r ic t ly decreasing.

(14.3) is still satisfied for any elliptic second order operator. Indeed :

f j(0)=aj+0(0) w h e n 0 4 0

Fj(c0)=aio+0((~ 2) w h e n co -~0

r t (co)=l Im(acaz) - t l2r t / to + 0(1) w h e n o 4 0 .

On the cont rary , (14.4) is not satisfied in any case (see example (14.7) below.

To be precise, we give a calcuius of the Fi(a)) :

(14.5) Lemma : Let a c C , a=cz+il3, cx,13~R and f3~O.

Let

We set :

Then we have :

6)

F(o~) = ~o acoS0-asino+ sin0cos0 dO ~0E[0,2~].

2 2 + ~

T(~o)=tan o) and U((o) - ~ )(~i tan o~ + T~ "

2

Re F(o~) = Z L o g u z ( o ) + I + Log" ~I13[ 2 T ( o ) ÷ I

f Ira F(00)= {

/

(Atan U(o) = Atan ~1

I (Atan U(~0) = Atan ~1

~j (Atan U(~0) - Atan ]~

) ~ ~ [o ,7 ]

+ ~) coE[~- ,~--1 3n

+ 2~) o e [ - - , 2 ~ ] . 2

112

To p r o v e t ha t , w e c h a n g e of v a r i a b l e s in t h e i n t e g r a l by s e t t i n g t = t a n 0

i n t e g r a t e a r a t i ona l f r a c t i o n by s epa ra t i ng real and i m a g i n a r y parts .

We deduce the f o l l o w i n g s t a t e m e n t :

and t h e n w e

(14.6) T h e o r e m : Let L be a second order operator, whose principal part in 0 is ."

[ D y - (~t + i[3t)Dx] [ D y - (¢22- i[32)D x]

with ~)'~2 > 0 and c~, cx 2 real. Let t o~ (0 ,x ,2x} . Then there exists a funct ion q(to) such that

F ° the Dirichlet problem for L on is [O,s]-regular in 0 i f and only / f s6 [0 ,q ( to ) [ . Here is a

formula for q(to) :

where

~(( .0 ) -- 2 ~ , Y ( o )

2 2 X (,~) * Y (,,~)

X ( t o ) : - - L o g I + ~_Log(~2+_____~2. 2 U2(to) +I cl2+[i 2 L~2J

2 i I

y ( t o ) :

with :

Q c~ i 2

A t a n U ( t o ) + A t a n U ( to ) - A t a n - - - A t a n - - , c0~[0 , - - ] 2

J 2 ~ 3 ~ 2 ~ + A t a n U (to) + A t a n U (to) - A t a n - - - A t a n to~ [~=,-~-]

l 2

c~ c, 3r~ ! 2

4 n + A t a n U ( t o ) + A t a n U ( t o ) - A t a n - - - A t a n - - , t o e [ ,2~[

1 2

2 2 (~ +[~

U : J J tan to + i

J

-1 P r o o f : W e h a v e seen t h a t q ( to )=2x l lm(F i -F2) ( to ) [. By se t t i ng

X(o~) = Re(F~-F2)(to)

Y(to) = Im(Fi-FE)(to)

and by using l emma (14.5) w e get t h e a b o v e s t a t emen t . •

(14.7) E x a m p l e : The case when ct l=c~z=0.

T h e n X(~/2)=Log([~l/[3 2) and Y 0 t / 2 ) = x . So :

113

r l ( z /2 ) = 1 [ ( l l n ) L o g ( f ~ l l ~ 2 ) l 2 + 1

Thus r l ( r t /2)< 1 if a n d o n l y if :

( 1 4 . 8 ) ILog (131/132)1 > ~.

On t h e o t h e r h a n d , X ( n ) = 0 e t Y(n)=2~t. So :

n(~) = 1.

So, w h e n (14.8) is sa t i s f ied , rl is n o t d e c r e a s i n g o n ]0,rt].

Here a r e a f e w t ab l e s of n u m e r i c a l v a l u e s f o r )1(60) w h e n el, ~2 =0 '

rtloJ

0.1

0,2

0,3

0,4

0,5

0.6

0.7

0.8

0,9

132

1

0,5

10

5 6.28

3.33 3.88

2.25 2.62

2 1.91

1.67 1.53

1.43 1.32

1.25 1.18

1.11 1.08

1

0.1

17.76

8.83

4,72

2.65

i ,30

1.26

1.20

1.12

1.06

1

e - ~

18.71

8.64

4.90

2.65

1

1.21

1.18

1,12

1.06

1

0.02

19.13

8.82

4,98

2,66

0,78

1,20

1,t7

1.1I

1,05

i

0.002

t - - 19,46 8.86

5,05

2.67

0.41

1.19

1.17

1.11

t .05

0.1

0.2

0.3

0.4

0.5

0,6

0.7

0.8

0,9

132

1 1

1 2

10 6,92

5 3.75

3.33 2.73

2.25 2.22

2 1.91

1.67 1.68

1.43 1.50 1.25 1.33

1.1I 1.16

1

10

2,56

t 35

1.48

1.36 1.30 1.27

1.261

1.264

1,259

e t t

1.58 I 1.06

1.18 1 0.85

1.06 I 0.79

t 1.01 t 0.77

1 I 0.78

1.01 I 0.81

1.04 I 0.85

1.09 [ 0.91

1,17 I 1,02

I

1 1

50 500

0.41

0.37

0.37

0.39

0.40

0,43

0.47

0.52

0.61

114

T h e n u m e r i c a l e x a m i n a t i o n s h o w s t h e f o l l o w i n g p h e n o m e n o n : t h e n u m b e r of local

e x t r e m a fo r rl a n d t h e v a l u e s of ~1 in t h o s e e x t r e m a d e p e n d o n l y on ILog b / b z l . More

p rec i se ly , t h e r e ex is t s B such t h a t :

• if ILog b / b 2 t ~ B , q is d e c r e a s i n g ;

. if ILog b / b J > B, )1 ad mi t s a r e l a t i v e m i n i m u m a n d a r e l a t i v e m a x i m u m . It is i n c r e a s i n g

b e t w e e n t h o s e e x t r e m a and d e c r e a s i n g ou t s ide .

We h a v e go t :

9 .405 < eB< 9 .4055 i.e. 2.24124 < B < 2.24130.

Here is a t ab l e g i v i n g t h e v a l u e s of t h e f u n c t i o n 3 in i ts e x t r e m a .

131 1

value [32 0.1

L I minimal 1.261

maximal 1.266

I

1

10

1.26t

1.266

1 1

exp(~t) exp(-rt)

I - - 1 1

1.215 1.215

1

50

0.775

1.200

1 1/20

0.02 1/1000

0.775 0.775

1.200 1.200

1

500

0.370

1.190

Here a r e t w o t a b l e s w h e r e w e i n d i c a t e t h e p o s i t i o n of t h e r e l a t i v e m i n i m u m and b

m a x i m u m . The f i r s t v a l u e w e c h o o s e fo r (13v13z) is (1,1/9.41), f o r w h i c h _ffL = 9.41, w h i c h is 2

n e a r t h e v a l u e w h e r e t h e r e l a t i v e e x t r e m a m e e t and v a n i s h .

131 1

t g/60 132 1/9.41

I minimum 0.542

maximum 0.545

t

13, 1

g/tO 132 lexp(-g)

minimum 0.500

maximum 0.604

1 1

1/9.42 119.43

0.540 0.539

0.546 0.547

1 1

1/50 1/500

. . . . . . .

0.498 0.499

0.614 0.621

1

t/9.44

0.539

0.548

1

1/9.45

0.538

0.548

1 1 i 10 expUt)

J 0.719 0.500

0.850 0.959 I

_ _ 1

1 1

1/9.5 1/10

0.535 0.526

0.550 0.562

1

5O

0.399

0.983

1

500

0.241

0.998

115

Let us end those c o m m e n t s abou t t h e case w h e n ~ = ~ z = 0 by a remark , We recal l

t h a t L is said [0,1]-regular in 0 if :

If u e t ~ l ( r ) is such t h a t LueL2(F) , w i t h u w i t h compact support , t h e n u~HH(F).

By compu t ing rl'(a), w e get t h a t ~ is decreas ing in t h e n e i g h b o r h o o d of m So, if L satisfies

(14.8), L is [O,1]-regular in 0 on F ~ f o r 6o nea r 0 or n, but is no t so for a~=n/2.

14.B Plane crack

Lemma (14.5) a l lows to compute :

Fl(2~)=2ir t and F2(2~)=-2i7: .

So, (14.2) yields :

X k = k / 2 .

W h e n k is odd, Z k is n o t in teger , and thus, if ~t=A k , L is no t in j ec t ive modulo polynomia ls

on S~(r z ~ ),

NOW, w e are i n t e r e s t e d in t h e case w h e n k=2(~ w i t h (~> 0, in teger . Then t he r ank of

JIe(A k) is i and its d e t e r m i n a n t has an order one zero in ~t k. S o , X ~ die(1) -l has an order

one pole in Xk' Thus, lemma (13.11) in fe r s t h a t ~ . ( 1 ) - ~ has a n order one pole in A k and t h a t

Ker ~ . ( l k) is one d imens ional . But, for X=A k :

dim P ~ ( r z ~ ) - d i m Q~-Z(F2#) = } . - ( I - i ) = i .

So, lemma (4.10) infers t h a t L is in j ec t ive modulo po lynomia l s on S~(F2~). We h a v e just got:

(14.9) P r o p o s i t i o n : L e t L be an homogeneous second order properly elliptic operator

with constant coefficients. Then, for Re A.>. O, L is injective modulo polynomials on S~'(F 2~ )

if and only if 2A is not an odd integer.

T h a n k s to (4.11) and (13.11), w e are able to calculate t he s ingular i t ies ar i s ing in I k for any

odd n u m b e r k : in t h a t case E~/P ~ =K ~ - cf (5.9). We get :

(14 .10) T h e o r e m : Let s > 0 be such that 2s is not an odd number. Let L be an

homogeneous second order properly elliptic operator with constant coefficients. Let

u e f f l l ( F 2n) be such that LueHS-~(F2~) .

Then, in the neighborhood of the bottom 0 of the crack, u splits into :

Uo+~o<k<2s c k rk/Hvk(0) k odd

116

where • u o belongs to HS+ i in the neighborhood of 0

F ( 0 ) . k / 2 F (0 ) . k /2 I 2

• v k ( 0 ) = e - e with the Fj defined in (13.9).

(14.11) Remark : If L=A, Fl(0)=i0 and F2(0)=-i0. We have vk(0 )=2 i sin kO/2. Our result

coincides w i t h Grisvard's one in [GR 3]. •

§15 F o u r t h order operators

According to (13.7), we wri.te L=I-Ij~j¢ 4 (Dy-a i D x) w i th Im a I, Im a2>0 and Im a 3,

Im a 4< 0. We are going to :

• de termine bases (Zj(9`)) of ~(9`) - cf 13.C,

• study the crack F 2~,

• study the hole F °.

We make all calculus in the two fol lowing si tuat ions :

(15 .1 ) all aj are distincts,

( 1 5 . 2 ) a l=a 2 and as=a 4.

We don ' t indica te proofs in the in te rmedia te case w h e n a t=a 2 and a3#a 4 (or the

reverse) .

15.A Basis of ~(9`)

(1) Case when the aj are distincts.

We set, for j=1,...,4, Hj(9`)=Kj(9`) -c f (13.10). We have indicated tha t this choice constitutes a

basis of ~Z(9,), except for a f in i te set of points 9,. Here, we precise that , by calculating the

w r onsk i an '~,d(9`) of the Hi(9`) i.e. the de te rminan t (in 0=0 for instance) of the matrix W(?,)

of the Hj and of their first three derivatives. We have the relat ions :

0 o Hj = 9`fjHj

O2oHj = (9`2f~ + 9`f'j)Hj

* 39`'f', +

On the other hand :

f'j' =2( f~ +fj)

fj(O)=aj.

So, W(9`) is the matrix, the columns of which are for j=l,...,4 :

(1, 9`aj, MX-I)a~=9`, 9`(9`-l)(9`-2)a~+9`(2-39`)aj).

Therefore ~ (9 ` ) is also the de te rminan t of the matrix whose columns are :

(I, 9`aj, 9`(9`-i)a~, 9`(9`-I)(9`=2)a~).

118

So, 0 is a t r i p l e roo t of ~ , 1 is a d o u b l e r o o t a n d 2 is a s impIe root . T h e r e is n o o t h e r root ,

fo r d°%d = 6.

W e a r e i n t e r e s t e d in t h e } .EC such t h a t R e } . ~ [ m - n / 2 , s + m - n / 2 ] , i.e. :

Re}.~ [1,s+ 1].

So, w e e x a m i n e t h e ae r a : Re}. )1 . Thus, w e h a v e to s t u d y e s p e c i a l l y t h e n e i g h b o r h o o d s of

2.=1 a n d }.=2 a n d o u r s t u d y w i l l be c o m p l e t e .

}.=1

T h e r a n k of W(1) is 2. Let (~,,...,,4) a n d (13,,...13 4) be a bas is of Ker W(1). W e set :

Z,= H, Z3=(}.-l)-I ~ c~} H}

Z2=H 2 Z 4 = ( } . - l ) - ' ~ 13jHj .

AS (~i s a t i s f y t h e r e l a t i o n s :

~j = 0 and ~" otjaj = 0,

t h e n t h e f i r s t t h r e e d e r i v a t i v e s of Z3(}.) ( w i t h r e s p e c t to O) a r e a n a l y t i c in 2,; so Z 3 i s

a n a l y t i c . T h e same is t r u e fo r Z 4.

The m a t r i x of t h e d e r i v a t i v e s of t h e Zj(1) is equa l to :

1 i

a I a 2

-1 -1

_ a I - a 2

0 0 )

o 0 I 2

Y- % a~ Y~j a i

-Y. ~i ai -Y~ Pi aj

To s t a t e t h a t i t s d e t e r m i n a n t is n o n zero, i t is e n o u g h to p r o v e t h a t fo r i ts s e c o n d d i agona l

b lock : l e t us suppose t h a t t h e r e ex i s t s ~, 13 such t h a t :

T h e r e f o r e :

~ ( ~ , j + 1 3 ~ j ) a ~ = 0 f o r k = 2 , 3 ,

But , b y d e f i n i t i o n of " i a n d ~ j , t h a t r e l a t i o n is s a t i s f i e d f o r k = 0 , 1 too . As t h e aj a re

d i s t i nc t s , w e d e d u c e t h a t :

~ i + N3 i = 0.

T h e r e f o r e , - - ~ = 0. So, t h e Zi(}.) f o r m a su i t ab l e bas is in t h e n e i g h b o r h o o d of }.=I.

119

7,=2

T h e r a n k of W ( 2 ) is 3. Le t (~v...,~4) be a bas is of i t s k e r n e l . We h a v e :

~ j ( a j ) k = 0 k=O,1,2.

We se t :

Zj = H i f o r j=1,2,3 a n d

T h e m a t r i x of t h e d e r i v a t i v e s of Zi(2) is :

Z4 = G - 2 ) -t 5~ ~i Hi"

i I 1 0

2aj 2a 2 2a 3 0

2 a ~ - 2 2 a ~ - 2 2 a ~ - 2 0

- 8 a I - 8 a 2 - 8 a 3 2 ~ c~ i a~

I ts d e t e r m i n a n t is n o n - z e r o . So, t h e Z i a re c o n v e n i e n t in t h e n e i g h b o r h o o d of 9`=2.

W e set , f o r j= 1,3 :

a n d

W e se t :

(2) Case when a~ = a 2 , a 3 = a 4 .

- Fj(8) e

g ( 0 ) = j a s i n e + cos0

J

I ' 0

Gi(O) = gj(~) d ~ .

H~ = e 9`Fj H 2 = G I e ~tFl

H 3 = ekF3 H 4 = G 3 J 'F3 .

T h o s e fou r f u n c t i o n s b e l o n g t o ~ ( ~ ) . Jus t l ike fo r case (I), w e p r o v e t h a t t h e w r o n s k i a n of

t h e Hj is a d = 4 p o l y n o m i a l t h e r o o t s of w h i c h a r e ~,=0 a n d ~.=2 ( s imple) , a n d ~,=I (double) .

• If 9`=i, t h e r a n k of W(1) is 2 a n d w e c o n s t r u c t Zv...~ 4 l ike in case (I).

• If 9,=2, t h e r a n k of W ( 2 ) is I a n d w e m a k e a g a i n t h e s ame c o n s t r u c t i o n t h a n in t h e case

w h e n aj a r e d i s t inc t s .

120

15.B Crack

In t h e cases (I) and (2), w e are going to ca lcu la te J~() . ) in t h e basis (Zj().)) w e h a v e

just c o n s t r u c t e d .

(1) Case when the aj are distincts.

If )`#I ,2, and w i t h t h e i n f o r m a t i o n - cf (14.5) :

F1(2n), F2(2n) = 2i~

Fz(2~), F4(2x) = - 2 i ~

w e get t h a t t h e co lumns of JIP().) a re :

(1 ,) .a j , e 2i~kc(j) " 21rtk~(j)x , ~.aje J w h e r e ~(I), ~(2)=I; ~(3),e(4) = - I .

Its d e t e r m i n a n t is equa l to :

2 -4 ) ` ( a4=a3) (a2-a ,) sinE(2~t)`).

It is z e r o if and o n l y if 2) .cZ.

We a re i n t e r e s t e d in t h e )`eC, Re) . ) I. So, let )`o be such a )., w i t h 2?toeN, T h e n )`o is a

double roo t of t h e d e t e r m i n a n t of dle().) and a s imple root of each of t he cofac to rs of JF()`).

M o r e o v e r , t h e r a n k of dle(), o) is 2. Thanks to (13.11), w e get :

~?..()`)-~ has a s imple pole in ) ` o

Ker ~£.()`o ) is 2 = d i m e n s i o n a l .

We deduce :

( i ) If ) . = k / 2 w i t h k odd, L is no t i n j e c t i v e modulo p o l y n o m i a l s on S~(F 2a) and :

EZ /P~(£ 2~) is 2 = d i m e n s i o n a l .

( i i ) If )` is a in teger , a n d ) ` >2, a s :

dim p~(r z') - dim Q~'-4(F2n) = ) . - I - ( ) . - 3 ) = 2

Thanks to (4.10), w e in fe r t h a t L is i n j e c t i v e modulo p o l y n o m i a l s on S~(F 2").

It r e m a i n s to s tudy t h e cases w h e n ) ,=I or ).=2,

I = 2

We recal l (cf 15 A,(1)) t h a t Z i = Hi fo r j=i ,2 ,3 and Z 4 = ( ) , -2 ) -~ ~ (~j H i .

As ~ ~i = 0, w e get t h a t :

(~i+(~2) e 2 i ~ +(~3+c~4) e - 2 i n ~ " = 2i(ct1+Ct 2) sin 2u)..

In t he same w a y , as ~ ~ j a j = 0, w e h a v e :

((~jaj+ ~2az) e 2i~" + (~3a3+ ct4a 4) e -2i~)" = 2i(ot)al+ (~zaz) s in2r[) , .

So, in t h e n e i g h o r h o o d of ) ,=2, t h e d e t e r m i n a n t of At'(),) is e q u i v a l e n t to t h e d e t e r m i n a n t :

121

1 1 1 0

al a2 a 3 0

2 i t~ ~. 2 i ~t 3. -2:i n ~. e e e

2 i t t l 2in~. - 2 i n k ale a2e a 3 e

w h i c h is equa l to :

,,(C~z+~21 s i n 2 x ( k - 2 )

I-2

, , (Ctla +c~2a2 ~ s in 2 n ( l - 2) ~ . - 2

s in 2it (k - 2) [(~j(a3-a )) + ctE(a3-a2)] 2i sin 2~t3..

I = 2

It has a s imple roo t in 3.=2. On the o t h e r hand, t he rank of d7'(2) is 3. So, t h a n k s to (13.11) :

~(3 . ) -~ has a s imple pole in 3.=2 ;

Ker ~-~(2) is l - d i m e n s i o n a l .

S2(F 2~ W i t h (4.10), w e deduce t h a t L is i n j e c t i v e modulo p o l y n o m i a l s on ).

}.=I

Just l ike in t h e case w h e n 3.=2, by using t h e basis Z i w e h a v e c o n s t r u c t e d in 15.A (i), w e

get t h a t t h e d e t e r m i n a n t of dF(3.) is e q u i v a l e n t in t h e n e i g h b o r h o o d of I = I to t he

d e t e r m i n a n t •

1 i 0 0

a I a 2 0 0

2 ig~ . 2i)%k e e (~1+C(2) s i n 2 a ( l - 1 ) ( ~ 1 + ~ 2 ) s i n 2 a ( X - l ) I-I .L-I

2i ~g 2i ~t~. s i a 2n (1- I ) , , t~)a)+c~2a2~ s in 2 n ( k - z ) ,vv,fRlal+~2a21 ale a2e 1 - 1 1 - I

w h i c h is equal to :

(a2_a,)Z ([5)c~ z _ ~2ct)) [ sin z#().- i ) ] z ~. - I

It is no t ze ro in 3.=I. So L is i n j e c t i v e modulo po lynomia l s on SI(F 2~ ).

As a c o n s e q u e n c e , for each 3.~N*, L is injective modulo polyornials on S ~ (F z~ ).

(2) Case when a ~ = a z, a3=a 4

For 3.~1,2, d7"(3.) is equal to t he f o l l o w i n g mat r ix ( w h e r e w e set xj :=Gj(2E)) :

1 0 1 0

)ta) 1 }.a 3 1 I

e x I e e x 3 e

a~e ~tajx1+ D e l a3e (3.a3x3 + l )e -2i ~ ]

122

Its d e t e r m i n a n t is equa l to :

4 sin22n;l. + X2(a l - a s ) 2 xlx 3.

If a3=~ j , t h e n x3--~ j and t h e d e t e r m i n a n t is equa l to :

4 s in22nX + ) t 2 ( a l - ~ ) 2 Jxtl 2 .

I ts d e t e r m i n a n t is n e c e s s a r i l y z e r o in e a c h i n t e g e r I > 2, b e c a u s e o f t h e d i f f e r e n c e of

d i m e n s i o n :

dim P~( r 2~) > dim Q~-4(F2~)

= cf (4.10). Thus, w e ge t :

xl=O.

In t h e same w a y x a=O. So :

de t d~(X) = 4 sin 2 2 n t .

Then , w h e n R e ~ ) 1 a n d t ~ 1 , 2, t h e c o n c l u s i o n s a re t h e s ame t h a n in t h e case w h e n t h e aj

a r e d i s t i nc t s .

}.=2

~ P ( 1 ) is equa l t o :

1

C~l+ C~3=0

2e~ la l+2 t~3a3 + 0~2+ C~4=0

o 1 o ]

;ta s 0

-2 in~ sin 2~t (~.- 2) e 2ic~ t

~ - 2

- 2 n l

Xa) I

2i n}. e 0

l a ) e 2in~" 2in}. e t a 3 e 2 i (~ la +c~z) sin 2t~(~.- 2) ~ - 2

Its d e t e r m i n a n t is equa l to :

sin 2~ (~.- 2) - 4 s in 2 n Z (Icqa i - X~la3+ c~e).

~ - 2

It has a s imple ze ro in I = 2 . The r a n k of J F ( 2 ) is 3. As b e f o r e , w e i n f e r t h a t L is i n j e c t i v e

modu lo p o l y n o m i a l s o n S2(F 2n).

}.=i

W i t h t h e Zj(X) w e h a v e c o n s t r u c t e d in 15.A, t h e d e t e r m i n a n t of dle(}.) is equal to :

Here, w e t a k e (cf 15,A) :

_ }-F 1 ~.F 1 ~ F 3 + ~-F 3 ) Z 4 ( 1 ) = ( 2 - 2 ) I((21 e + oz G~ e + ~z e a 4 G 3 e

w h e r e t h e c~j sa t i s fy :

123

It is n o n zero in ~=1.

4 [ s i n 2 ~ O . - , ) ]2 (C~213j_cqf~z)" A - 1

The syn the s i s of all t h a t y ie lds :

(15.3) P r o p o s i t i o n : Let L be a four th order homogeneous proper ell iptic operator with

constant coefficients . Let 2(. e C, Re2t>~ 1. Then L is injective modulo polynomials on S~'(£ 2~ )

i f and only i f 2). is not an odd integer.

(15.4) T h e o r e m : Let L be as in (15.3) and s>~ 0 such that 2s is not an odd number. Let

u e ~1 a (i. 2 • ) such that L u e H s - 2 (£ 2 ~ ). Then, in the neighborhood o f O, u splits into :

U=Uo+~2<k<2~ ~,~i~2 %,i rk/2 vk,i(°) k odd

where u o belongs to H s+ z in the neighborhood o f 0. Here are the formulae for the vk, i :

(1) i f the a i are distincts :

Vk.j(O) = ~ i~e~4 t~.j e Fe(O)'k/2

where the (c~).j ..... o~4, j) are, f o r j= l , 2 two independant solutions o f the system :

(2) i f a j = a 2 and

where the

~ j = 0

Y ~iaj = O.

a 3 = a 4 F . k12

I V k , j = C~I, j e

F . k / 2 F , k / 2 F , k / 2 t 3 3

+ ~2, i GI e + ~3, j e + ~4 , j G3 e

(~'.i ..... c~4. i) are, for j= l , 2 two independant solutions o f the system :

~1+~3= 0

(oJal + °~3a3)2- + ~2 + °~4 = 0.

(15 .5) R e m a r k : If L=A z, w e h a v e : a l=i , a 3 = - i and :

Fl(0)=i0, F3(0)=- ie , G j ( O ) = ( 1 - e - 2 i ° ) / ( 2 i ) , G 3 ( 0 ) = ( e 2 i ° - l ) / ( 2 i ) .

As s o l u t i o n s (~t.j,...,~4,j) of the sys tem :

(~1+(~3 = 0

[ . ' ( ~ l - ~ 3 ) t - i + c~ 2 + ~ 4 = 0

w e m a y choose t h e complex n u m b e r s :

Cll,l = 0 (~2j =i Cl3,1 = 0 (~4j= -i

124

So, w e deduce tha t :

(Xl.2=i ~2.2 = k / 2 C~3.z = - i ~4.2= k / 2 .

k ~ -2)0 V~,t(O) = COS ~0 - COS(

Vk,2(O) ( ~ - 2 ) ' k k . = sm~O - 7 s m ( 7 - 2 ) 0 .

By se t t ing k = 2 8 + I , we get t he same singular funct ions than those g iven by Grisvard in

[GR 3]. •

I5.C Hole : r°=R2k(O}

At first, he re is a general s ta tement :

(15.6) Propos i t ion : Let f2* be a smooth bounded domain in R 2, containing O. f2=O* \{0}.

Let L be a strongly elliptic 2m-order operator. Then for each s > m - 1 ,

L : t~tmt~H s+ m(~) ~ Hs-m(~)

is a Fredholm operator and its index is =m(m= 1) /2 .

Proof : For s > m - 1 , L acting on t~ImnH s+ re(f2) is a res t r ic t ion of L acting on ~ m n H s + re(O*),

ffImC~H s+ re(O) being indent i f ied to the subspace of the u belonging to fftmNH s+ re(f2* ) such

tha t Dau(0)=0, I~l,<m-2. By adding to L a suitable constant , we may suppose tha t L is an

isomorphism from ~tm(O * ) onto H- m(f~*). So, L is an isomorphism from ~mnHm+ s(O*) onto

H s- re(O*). As the codimension of ffimNH m+ s(f2) in I~m ¢~H m+ s(k~*) is re(m-i )~2, and as

H s- re(f2) may be ident i f ied to H s- m (~*) we deduce the proposition. •

So, if m=2 , the index is - i and w e may expect a unique singular func t ion w h e n s>l . In

the fo l lowing, we on ly suppose tha t L is proper el l iptic and w e are going to construct that

singular func t ion wh ich arises in X=2. Then we wil l p rove that it is unique.

(1) Case when the aj are distincts.

According to 15.A, the Zj are the Hj for j=1,2,3 and Z4= ( ~ - 2 ) -t ~ ~i Hi"

The first th ree columns of .~e(Z) are, for j=1,2,3 :

(e 2in~'t (j) - I)

(e 2in~'c (j) - l ) a j

(e 2in~'t (j) = 1)(a~ (~-l)-aj)

(e 2in~'c(j) =1)(a~ Gt- I)(X=2)+ (2 - 3;t)a i)

125

w h e r e E(j)=I for j=1,2 and E(3)=- I .

Thanks to the re la t ions ~ k • ~ j a j =0, k=0,1,2, we p rove t ha t the four th column may be

w r i t t e n in the form, w i t h

s(2,):=(sin 2r~;()(2,-2) -t :

s(2,) ~ t<j~2 cxi

s(2,) X )(j(20tj aj

2 S(2,) El<j< 2 (2,-I) C~j aj - c~j aj

s(;() Xt<j~ 2 ( 2 - 3 ; ( ) c ~ j a j + ( ; ( - l ) [ ~ , ~ j ~ 2 clj a~ ( e 2 i ' ~ ' - l ) + ~ ' , ~ j ~ 4 C~i a~ ( e - ' i ' Z - l ) ] / 2 i .

Let ~P(?~) be the matr ix :

i 0 0

0 i 0

0 i i

0 (2 -32`) -I 0

0

0

0

1J Then dg'o:=~Pd)~ enjoys the same proper t ies than dr( ; ( ) in the ne ighborhood of ;(=2

for ~P(2) is enver t ib le .

At first , we not ice t ha t the rank of dFo(2`) isl. On the o ther hand, dleo(;() may be

wr i t t en in the form of the fo l lowing product :

g$(;() o%(2`) ~(2`) = ~ o ( Z )

w h e r e g~(;() and ~(;() are diagonal matr ices :

the diagonal of ~( ; ( ) is (i, I, (;(=i), (2 , - i ) ( ; ( -2))

the diagonal of ~O(2,) is(eZi"z=1, e2i~Z-l , e - 2 i ~ z - 1 , i)

and owL(;() may itself be w r i t t e n in the form :

dVto(;() ~o ( ; ( ) - I

w h e r e ~Po(;() is the fo l lowing inver t ib le matrix in the ne ighborhood of ;(=2

I

0

0

0

and dVLo(;() is the matr ix :

0 0 - s(~.) el,

1 0 - s ( 3 . ) c~ 2

0 1 0

0 0 1

126

w h e r e :

1 1 I 0

a ~ a 2 a a 0 2 2 2

a I a z aa 0

3 3 3 a ) a z a 3 C44(/1.)

- 2 i,x ;I

2iC44(X) = ~ l < j < 4 u i a ~ e l K - 2

As ~ ~j a~ .O , t h e n C 4 4 ( 2 ) * 0 a n d 8 'Lo(2) is i n v e r t i b l e .

In t h e n e i g h b o r h o o d of ) `=2 w e h a v e -

aleo(~ ) -i = ~(X)-1 g~L(~) -i a~(~) - I

As e + 2 i = z - I h a s a s imp le ze ro in ) ,=2 , by d e n o t i n g mjk(~) t h e c o e f f i c i e n t s of o%(),) -I and

~jk(~t) t h o s e of ag 'o(X)-1 w e d e d u c e t h a t :

-2 -~jk(),) .- ( ) , - 2 ) ~ jk ( ) j=1,2,3 a n d k = 4

~ j t ( ~ ) - ( ~ - 2 ) -I ~ j k ( ~ ) ( j , 4 a n d k~e4) or ( j = 4 , k = 4 )

r , jk(),) - ~ jk(X) j=4 a n d k # 4 .

We see t h a t ~z .4 (2 ) a n d ~x2.4(2) a re n o n zero.

As a c o n s e q u e n c e of all t h a t , w e ge t t h a t ) `=2 is a d o u b l e po le of Jleo(),) -~ a n d t h a t :

( 1 5 . 7 ) l im z _,~ ( X - 2 ) -I ~qeo(~,)-s ha s i ts r a n k equa l to I.

T h e r e f o r e , ~qe(~.)-1 h a s t h e same p r o p e r t i e s , and such is t h e case fo r ~ ( ~ t ) - ' . T h e r e f o r e L is

n o t i n j e c t i v e m o d u l o p o l y n o m i a l s o n S2(F°) . As Q~'- 4 is r e d u c e d to {0}, w e i n f e r f r o m (15.7)

t h a t t h e space EE(F°,L) = cf (3.8) - is g e n e r a t e d b y a s i ngu l a r f u n c t i o n in t h e f o r m :

rZ(v(0) Log r + w ( 0 ) )

a n d b y o t h e r f u n c t i o n s in t h e f o r m :

r2vk(0) .

2 Let us p r o v e t h a t r v~(0) is p o l y n o m i a l . We h a v e :

L(r 2 v k) = 0.

T h e r e f o r e :

~£(2) v~ : 0.

But, t h e k e r n e l of ~I('(2) is 3 - d i m e n s i o n a l , a n d such is t h e case fo r gg(2) t h a n k s to (13.11).

Thus t h e r e ex i s t s e x a c t l y 3 i n d e p e n d a n t v k . But t h e d i m e n s i o n of p 2 ( r ° ) is 3 a n d LP2=0.

2 2 So, r v k e P a n d t h e d i m e n s i o n o f E Z / P a i s l . •

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(2) if a~=a 2 a n d a 3 = a 4 , w e ge t t h e s ame r e su l t s b y s im i l a r a r g u m e n t s ,

Now, w e l ook a t t h e case w h e n Re~.> 1 a n d ~.m2. I t is c l ea r t h a t :

de t JIP(~.) = ( e 2 i n ~ - l ) 2 ( e - 2 i ~ - l ) 2 ~ ( ~ . )

w h e r e ¢f.d(~.) is t h e w r o n s k i a n of t h e Hj(~t),

In ou r ae ra , t h a t d e t e r m i n a n t is z e ro if a n d o n l y if ~. is i> 3 a n d i n t e g e r . Such a ;t is a f o u r t h

o r d e r ze ro of d e t dlP(~,) a n d a t h i r d o r d e r ze ro of al l c o f a c t o r s of AP(Z), So /£.-~(?.) ha s a

s imple po le a n d t h e k e r n e l of 2L(~.) is 4 - d i m e n s i o n a l . But :

d im P~'( 1 "°) - d im Q~. -4(Fo) = ~ t + l - ( ~ - 3 ) = 4.

Thus L is i n j e c t i v e m o d u l o p o l y n o m i a l s on S ~ ( F ° ) ,

F ina l ly , w e ge t :

(15 ,8) T h e o r e m : Let L be a p r o p e r e l l ip t ic f o u r t h o r d e r h o m o g e n e o u s o p e r a t o r wi th

constant coef f ic ients . Let s ~ l . Let u ~ I Z ( R 2 \ { 0 } ) be such that L u s H s - 2. Then :

, i f s < 1, u ~ H s + 2 in the ne ighborhood o f O.

• if s > l ,

U=Uo+ c r 2 [ v ( 0 ) Log r + w(0 ) ]

with • Uo~ H s+ 2in a ne ighborhood o f O, c ~ C .

• r a n d w do not depend on u and i f L = A 2 we have :

v = l and w = 0 .

(15.9) R e m a r k : r H v ( 0 ) ~ P 2 ( F ° ) : i t is p o l y n o m i a l . •

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